I think what is bothering you is that not that you can solve this equation for some x but to solve it for all x.

You begin by finding one solution of,
You can do this by using arcsine, but you should know what this angle is,
and
These are all the non-coterminal angles, thus, all solutions are,

The problem which many people have is finding what the angle has to be. Notice that if the problem was then the angle is you should have this memorized, but the problem asks for thus, one such solution is, . Now find where this is located the on circle and in which quadrant. This is located in 4th quadrant producing angle of . How do you know what this angle is? Well, you now it is backwards and the full circle has thus if you subtract them properly you get . To find the other non-coterminal solution you now since it is negative it is in the 3rd quadrant. Thus, it is below the axis. Thus, a total of if you add them properly. Now since adding or subtracting it does not change the sines values they are all solutions thus, this is what is doing there.
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There is a much easier way. Look at this: for
Then ALL the solutions for are,
Where is the inverse sine function.

Thus, in your problem you get,
But
Thus,
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You might be thinking then what is the difference between the first solution I gave and this one? The answer is there is no difference, the second one is a fancier way of expressing all the solutions for . In fact, if were to write them out you will see you get the exact same thing.